Fig. 9 shows the curve $y = f(x)$, where $f(x) = \frac{1}{\cos^2 x}$, $-\frac{1}{2}\pi < x < \frac{1}{2}\pi$, together with its asymptotes $x = \frac{1}{2}\pi$ and $x = -\frac{1}{2}\pi$.

\includegraphics{figure_9}

\begin{enumerate}[label=(\roman*)]
\item Use the quotient rule to show that the derivative of $\frac{\sin x}{\cos x}$ is $\frac{1}{\cos^2 x}$.
[3]

\item Find the area bounded by the curve $y = f(x)$, the $x$-axis, the $y$-axis and the line $x = \frac{1}{4}\pi$.
[3]
\end{enumerate}

The function $g(x)$ is defined by $g(x) = \frac{1}{2}f(x + \frac{1}{4}\pi)$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Verify that the curves $y = f(x)$ and $y = g(x)$ cross at $(0, 1)$.
[3]

\item State a sequence of two transformations such that the curve $y = f(x)$ is mapped to the curve $y = g(x)$.

On the copy of Fig. 9, sketch the curve $y = g(x)$, indicating clearly the coordinates of the minimum point and the equations of the asymptotes to the curve.
[8]

\item Use your result from part (ii) to write down the area bounded by the curve $y = g(x)$, the $x$-axis, the $y$-axis and the line $x = -\frac{1}{4}\pi$.
[1]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C3 2011 Q9 [18]}}